Welcome to the 23rd Carnival of Mathematics: Haiku Edition! First, I must apologize for the delay: I usually have very little trouble with my hosting provider, but of course it went down just when the CoM was supposed to be posted. But it’s free, so I can’t complain! It’s back up now, and will hopefully stay that way.

For this edition of the CoM, I decided to write a short seventeen-syllable haiku about each of the excellent seventeen submissions I received (along with additional commentary of the more prosaic variety). I’ve arranged the posts more or less in order of required mathematical background, but don’t stop halfway through because then you’ll miss the pretty pictures at the end. Enjoy!

JD2718 shares a gem of a puzzle involving the sum of some angles. It’s tricky—are you up to the challenge? I would especially encourage would-be solvers to come up with a nice geometric solution (I couldn’t)!

Pascal’s Triangle:
writing it out is a chore.
How fast does it grow?

Foxy, of FoxMaths! fame, presents an interesting two-partanalysis of the asymptotic growth of the rows of Pascal’s triangle—not the growth of the actual values in the rows, but of the space needed to write them!—making use of some clever algebraic gymnastics and asymptotic analysis.

In how many ways
can the Nauru graph be drawn?
The answer: a lot!

David Eppstein of 0xDE presents The many faces of the Nauru graph: a collection of diverse ways to visualize a particular graph which he dubs the “Nauru graph”, due to the similarity of one of its drawings to the flag of Nauru. Planar tesselation, hyperbolic tesselation, embedding on the surface of a torus… all that and much more, with, yes, pretty pictures for everything! Even those who don’t understand the article itself should still go take a look, solely for the sake of the pictures. =)

Thanks to everyone for the great submissions, I had a fun time reading them and putting this together. The next CoM will be hosted at Ars Mathematica. As always, email Alon Levy (including “Carnival of Mathematics” in the subject line) if you’d like to host an edition.

On point 15, the angle puzzle, there is a nice geometric solution, though the analytic one with complex numbers might be easier to find. I don’t remember where I’ve read it, maybe it was an IMO task or something. Anyway, it’s this.

As in the task, define the points A=(0,0), D=(3,0), E=(3,1), F=(2, 1), G=(1, 1). Also, take the point T=(2,-1). Then, segment AT is equal to segment ET and they are perpendicular, so the triangle ATE is an isosceles right triangle. Thus, the TAE angle is equal to the TEA angle. Also, obviously, the DAT angle is equal to the DAF angle, so the TAE angle is equal to the DAE angle plus the DAF angle. Finally, the DAG angle is obviously half a right angle.